reserve X for ARS, a,b,c,u,v,w,x,y,z for Element of X;
reserve i,j,k for Element of ARS_01;
reserve l,m,n for Element of ARS_02;
reserve A for set;
reserve
  S for Group-like quasi_total partial invariant non empty non-empty TRSStr;
reserve a,b,c for Element of S;
reserve S for Group-like quasi_total partial non empty non-empty TRSStr;
reserve a,b,c for Element of S;
reserve
  S for Group-like quasi_total partial invariant non empty non-empty TRSStr,
  a,b,c for Element of S;

theorem
  S is (R2) (R4) implies (a")" * 1.S <<>> a
  proof
    assume
A1: S is (R2) (R4);
    take (a")" * (a" * a);
    a" * a ==> 1.S by A1;
    hence (a")" * (a" * a) =*=> (a")" * 1.S by Th2,ThI3;
    thus (a")" * (a" * a) =*=> a by A1,Th2;
  end;
